Quasi-coherent Hecke category and Demazure Descent

TL;DR

This paper introduces the quasi-coherent Hecke category for reductive groups, constructs its monoidal action on derived categories, and proves an equivalence with G-equivariant sheaves via Demazure Descent Data.

Contribution

It defines the quasi-coherent Hecke category and establishes a new equivalence between descent categories and G-equivariant sheaves.

Findings

The Hecke category acts monoidally on derived categories of B-equivariant sheaves.

Demazure Descent Data induces an equivalence of categories.

The framework generalizes classical descent results to a quasi-coherent setting.

Abstract

Let G be a reductive algebraic group with a Borel subgroup B. We define the quasi-coherent Hecke category for the pair (G,B). For any regular Noetherian G-scheme X we construct a monoidal action of the Hecke category on the derived category of B-equivariant quasi-coherent sheaves on X. Using the action we define the Demazure Descent Data on the latter category and prove that the Descent category is equivalent to the derived category of G-equivariant sheaves on X.